Many techniques and systems have been developed to image the interior structure of a turbid medium through the measurement of energy that becomes scattered upon being introduced into a medium. Typically, a system for imaging based on scattered energy detection includes a source for directing energy into a target medium and a plurality of detectors for measuring the intensity of the scattered energy exiting the target medium at various locations with respect to the source. Based on the measured intensity of the energy exiting the target medium, it is possible to reconstruct an image representing the cross-sectional scattering and/or absorption properties of the target. Exemplary methods and systems are disclosed in Barbour et al., U.S. Pat. No. 5,137,355, entitled “Method of Imaging a Random Medium,” (hereinafter the “Barbour '355 patent”), Barbour, U.S. Pat. No. 6,081,322, entitled “NIR Clinical Opti-Scan System,” (hereinafter the “Barbour '322 patent”), the Barbour 4147PC1 application, and the Barbour 4147PC2 application.
Imaging techniques based on the detection of scattered energy are capable of measuring the internal absorption, scattering and other properties of a medium using sources whose penetrating energy is highly scattered by the medium. Accordingly, these techniques permit the use of wavelengths and types of energy not suitable for familiar transmission imaging techniques. Thus they have great potential for detecting properties of media that are not accessible to traditional energy sources used for transmission imaging techniques. For example, one flourishing application of imaging in scattering media is in the field of optical tomography. Optical tomography permits the use of near infrared energy as an imaging source. Near infrared energy is highly scattered by human tissue and is therefore an unsuitable source for transmission imaging in human tissue. However, these properties make it a superior imaging source for scattering imaging techniques. The ability to use near infrared energy as an imaging source is of particular interest in clinical medicine because it is exceptionally responsive to blood volume and blood oxygenation levels, thus having great potential for detecting cardiovascular disease, tumors and other disease states.
A common approach for the reconstruction of an image of the cross-sectional properties of a scattering medium is to solve a perturbation equation based on the radiation transport equation. The radiation transport equation is a mathematical expression describing the propagation of energy through a scattering medium. The perturbation formulation relates the difference between coefficient values of the true target and a specified reference medium, weighted by a proportionality coefficient whose value depends on, among other things, the source/detector configuration and the optical properties of the medium. In practice, tomographic measurements consider some array of measurement data, thus forming a system of linear equations having the formu−ur=δu=Wrδx,  (1)where δu is the vector of differences between a set of measured light intensities (u) and those predicted for a selected reference medium (ur), Wr is the Jacobian operator, and δx is the position-dependent difference between one or more optical properties of the target and reference media (i.e., a change in absorption coefficient δμa, a change in the reduced scattering coefficient, μ′s, or, in the diffusion approximation, the diffusion coefficient δD, where D=1/[3(μa+μ′s)]). The operator, referred to as the weight matrix, has coefficient values that physically represent the fractional change in light intensity at the surface caused by an incremental change in the optical properties at a specified point in the medium. Mathematically this is represented by the partial differential operator ∂ui/∂xj, where i is related to the ith source/detector pairs at the surface of the medium, and j to the jth pixel or element in the medium.
Although the perturbation equation in Eq. (1) can be solved using any of a number of available inversion schemes, practical experience has shown that the accuracy and reliability of the results obtained can vary greatly due to uncertainties and errors associated with the quality of the measurement data, inaccuracies in the physical model describing light propagation in tissue, specification of an insufficiently accurate reference state, the existence of an inherently underdetermined state caused by insufficiently dense measurement sets, weak spatial gradients in the weight function, and so forth.
In practice, a matter of considerable concern is the accuracy with which the reference medium can be chosen. An accurate reference is one that closely matches the external geometry of the target medium, has the same size, nearly the same internal composition, and for which the locations of the measuring probes and their efficiency coincide well with those used in the actual measurements. While such conditions may be easily met in numerical and perhaps laboratory phantom studies, they represent a much greater challenge in the case of tissue studies. Confounding factors include the plasticity of tissue (it deforms upon probe contact), its mainly arbitrary external geometry and internal composition and the considerable uncertainty stemming from the expected variable coupling efficiency of light at the tissue surface. The influence of these uncertainties can be appreciated when it is recognized that the input data vector for the standard perturbation formulation (i.e., Eq. (1)) is actually the difference between a measured and a computed quantity. This vector contains information regarding the subsurface properties of the target medium that, in principle, can be extracted provided an accurate reference medium is available.
In practice, however, there are two significant concerns that are frequently encountered in experimental studies and are not easily resolvable especially in the case of tissue studies. One concern is the expected variable coupling efficiency of light entering and exiting tissue. Nonuniformity in the tissue surface, the presence of hair or other blemishes, its variable deformation upon contact with optical fibers, the expected variable reactivity of the vasculature in the vicinity of the measuring probe all serve to limit the ability to accurately determine the in-coupling and out-coupling efficiencies of the penetrating energy. Consideration of this issue is critical as variations in the coupling efficiency will be interpreted by the reconstruction methods as variations in properties of the target medium and can introduce gross distortions in the recovered images. In principle, the noted concern can be minimized by adopting absolute calibration schemes, however, in practice the variability in tissue surface qualities will limit reliability and stability of these efforts.
A second concern stems from the underlying physics of energy transport in highly scattering media. One effect of scattering is to greatly increase the pathlength of the propagating energy. Small changes in the estimated absorption or scattering properties of the medium can, depending on the distance separating the source and detector, greatly influence the density of emerging energy. This consideration has important implications regarding the required accuracy by which the reference medium must be specified. In the context of perturbation formulations, the reference medium serves to provide estimates of the predicted energy density as well as to provide the needed weight functions that serve as the imaging operators. The difficulty is that the computed reference intensity values are extremely dependent on the optical coefficient values of the reference medium. Significantly, this dependence is a nonlinear function of the distance between source and detector. It follows that a small change in the optical properties of the reference medium may influence the value of the computed intensity differences (δu) by a relative amount that may be significantly different for each source/detector pair, thereby altering the information content of the data vectors. This can lead to the recovery of grossly corrupted images. Whereas, in principle, such effects may be overcome by use of recursive solutions to the perturbation equation (i.e., Newton-type updates), in practice this can require extensive computational efforts, especially in the case of 3D solutions. Moreover, it is well appreciated that such efforts to improve on first order solutions to the perturbation equation (e.g., Born or Rytov solutions), can fail if the initial estimate chosen for the reference medium is insufficiently accurate.
One alternative to devising absolute calibration schemes is to devise methodologies whose solutions are intrinsically less sensitive, or better still, do not require such information, but nevertheless are capable of providing accurate descriptions of certain features of highly scattering media. While a range of empirical methodologies can be devised, it is desirable that they be broadly extendable without requiring undue physical approximations, since these are generally incompatible with model-based methods.
An approach previously adopted is to directly relate relative detector readings, obtained from comparison of detector values derived from two different target media (usually media with and without the included object), to the weight matrix computed based on a previously assigned reference medium. R. L. Barbour, H. Graber, R. Aronson, and J. Lubowsky, “Model for 3-D optical imaging of tissue,” Int. Geosci. and Remote Sensing Symp., (IGARSS), 2, 1395–1399 (1990). While capable of producing good quality images of internal structure of a target medium, the method proved to have limited utility as it did not produce solutions having physical units, thereby rendering specific interpretation difficult, as well as limiting efforts to compute recursive solutions.
For the forgoing reasons, there is a need for image reconstruction techniques based on the detection of scattered energy that (1) do not require absolute calibration of, and absolute measurements by, the detectors and other elements of the apparatus, (2) make the standard perturbation equation less susceptible to variations between boundary conditions and properties of the reference medium and the target medium, and (3) produce solutions having physical units.